When Max Hollein, the newly appointed director of the Metropolitan Museum of Art in New York, first sits down in his Fifth Avenue office, he will have thousands of problems to address. The Met is America’s greatest museum, after all: It has a collection of nearly 2 million objects, 7 million annual visitors, a budget of $305 million and a staff of 2,200. And it has endured a turbulent few years.

But Hollein, 48, who will be just the 10th director at the Met in 148 years, will have two first-order issues to confront. One will be: How much energy should the Met put into modern and contemporary art? The second will be: Can I work with this guy?

“This guy” is Daniel H. Weiss, the Met’s president and CEO. He has been running the Met since the surprise resignation last year after eight years in the job of Englishman Thomas Campbell. When Hollein joins the Met, he will share responsibility for running the institution with Weiss in a new power-sharing arrangement that may prove difficult to manage.

Similar arrangements have proved dysfunctional at the Getty and at the Guggenheim, leading to premature departures by high-profile directors who felt interfered with or undermined.

The leadership-by-committee model is in stark contrast to the Met’s hierarchy under Campbell and his predecessor, Philippe de Montebello, who ran the Met like a semi-divine sovereign.

Hollein was born in Vienna and is the son of the Pritzker Prize-winning architect Hans Hollein. He comes to the Met after only two years running the Fine Arts Museums of San Francisco (comprising the de Young and the Legion of Honor museums).

Before that, he was the long-term director of a trio of museums in Frankfurt: the Städel Museum, the Schirn Kunsthalle and the Liebieghaus sculpture collection. His first museum job was at the Guggenheim, as chief of staff and executive assistant to its expansion-minded director, Thomas Krens. He has a strong track record in digital innovation and in finding innovative ways to expand and diversify audiences.

Both Weiss and Hollein are art historians with MBAs. Hollein, who will focus on the “content” side of the museum, will report to Weiss, who will focus on the “administrative” side of things.

But who will explain to the Met’s staff, and to its supporters, where the line is between content and administration?

Both have a reputation for reasonableness and equanimity. But having two leaders can be difficult.

The bigger issue — how many resources should the Met devote to modern and contemporary art? — is more existential.

One argument is that the Met, in opening the Met Breuer and planning a new $600 million wing for modern and contemporary art, overreached. These plans not only led to worrying deficits, which in turn led to staff reductions and other forced economies; they also lured the Met away from its true mission, which is to be an encyclopedic museum devoted to art from all over the world and every period.

Why try to do contemporary art when New York already has so many museums (the Museum of Modern Art, the Whitney Museum of American Art, the Guggenheim and the New Museum, to name the most obvious) doing just that?

One answer — because the wealthy board members who keep the Met running are all besotted with contemporary art and make it a condition of their support — may be partially true. It is something Hollein, who has track records with both contemporary and historical art, should try to resist. But it’s not the full story.

The fact is that the Met should be showing and collecting contemporary art. If it doesn’t, it will seem less vital and relevant to contemporary audiences. It will also miss out on the many opportunities provided by living artists and recent art to shed light on its historical collections. Even de Montebello — fundamentally a conservative man as well as a beloved director — was on board with this.

But Hollein will need to balance the urge to show contemporary art with all of the Met’s other core commitments. That may prove difficult when so many people on the board — many of them with collections of modern and contemporary art they may one day want to give to the Met — are advocating for their own interests.

The Met’s real interests are not the same as the interests of its wealthy donors. The Met has a broader and deeper mission. To any sane outside observer, spending $600 million on a wing for modern and contemporary art seems unnecessary when aspects of those core interests are in jeopardy.